ZeroNim

ZeroNim is a variant of the impartial game Nim. In standard Nim, players take turns removing objects from distinct piles. The player who takes the last object wins. ZeroNim modifies this rule. In ZeroNim, the player who takes the last object loses. This is an example of a misere play game.

Like standard Nim, ZeroNim can be analyzed using the Sprague-Grundy theorem. The nim-sum of the pile sizes determines the winning strategy. The nim-sum is calculated by taking the bitwise XOR of the sizes of all piles. A position in Nim (and ZeroNim) is a losing position (P-position) if its nim-sum is zero, and a winning position (N-position) if its nim-sum is non-zero.

However, the misere play condition in ZeroNim introduces a slight complication. The standard Sprague-Grundy theorem applies directly to normal play games where the last player to move wins. For misere play, the winning strategy is the same as normal play until a specific point. Specifically, if the game reaches a state where all piles have size 1, the winning strategy flips.

If a player is faced with a position where the nim-sum is non-zero, they can always make a move to a position with a nim-sum of zero. If a player is faced with a position where the nim-sum is zero, any move they make will result in a position with a non-zero nim-sum. In ZeroNim, the goal is to force the opponent into a position where they are forced to take the last object. This means leaving the opponent with a nim-sum of zero when they cannot avoid taking the last object is a losing move for the player who made it.